A mathematical group is defined as a set and an operation satisfying certain rules. Set S be a set and f a binary function from S×S to S (f:S×S→S). {S, f} is a group if f is such that

• The Associative Rule Holds: f(a,f(b,c)) = f(f(a,b),c) for all a,b,c in S.
• There is an Identity: There exists an element e of S such that f(e,a)=a for all a in S.
• There are Inverses: For all a in S there exist an element b such that f(b,a)=e, the identity element of S.

The nature of the elements of S is unimportant. The elements are just identifiers. Any set with the same number of elements at S could equally well be used to define the group. The nature of elements is of significance only as a convenient was to define the group operation f.

The group operation is most conveniently defined in terms of an operation table. For example,

The group is equivalent to the addition of integers modulo 3; the sum of two numbers is replaced by the remainder when the sum is divided by 3. The identity element e corresponds to 0. The inverse of 1 is 2 because 1+2=3 and hence (1+2)(mod 3) = 0. Likewise the inverse of 2 is 1. Zero is its own inverse.

A group is thus a generalization of arithmentic. There is no requirement of symmetry; i.e., that f(a,b)=f(b,a). A group with an operation that is symmetric is called an abelian group.

Groups are often most conveniently defined in terms of relations for generating elements. For material on this topic see Groups, Generators, Relations and Cayley Diagrams.